Semi-Classical Asymptotics of Bergman and Spectral Kernels for (0,q)-forms

In this paper, we develop a new scaling method to study spectral and Bergman kernels for the k-th tensor power of a line bundle over a complex manifold under local spectral gap condition. In particular, we establish a simple proof of the pointwise asymptotics of spectral and Bergman kernels. As a new result, in the function case, we obtain the leading term of Bergman kernel under spectral gap with exponential decay. Moreover, in the general cases of (0,q)-forms, the asymptotics remain valid while the curvature of the line bundle is degenerate.